A-level · Integration

Areas with integrationExam-style questions

Setting up a definite integral for each kind of area question. A region between a curve and the x-axis, with the limits found from the x-intercepts; a region below the axis, where the integral is negative and the area is its size; a curve that crosses the axis inside the range, split at the crossing; the region between a curve and a line, as top minus bottom between the points where they meet; and regions made of a triangle and a curved piece.

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29 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
Figure 1In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Figure 1 shows a sketch of part of the curve $C$ with equation

$y = 8\sqrt{x} - 3x \qquad x \geqslant 0$

The point $A$ is the maximum point of $C$ .
(a) Find, using calculus, the coordinates of $A$ .

(5 marks)

The curve passes through the origin $O$ and meets the positive $x$ -axis at the point $B$ .
(b) Using algebra, find the $x$ coordinate of $B$ .

(2 marks)

The region $R$ , shown shaded in Figure 1, is bounded by the line segment $OA$ , the curve $C$ between $A$ and $B$ , and the $x$ -axis.
(c) Use algebraic integration to find the area of $R$ , giving your answer to 3 significant figures.

(5 marks)
Level 5 · 12 marksTry it
2
Figure 1In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Figure 1 shows a sketch of part of the curve $C$ with equation

$y = x^2 - 2x^{\frac{3}{2}} + 10 \qquad x \geqslant 0$
(a) Use calculus to find the range of values of $x$ for which $y$ is decreasing.

(4 marks)

The point $P(4, 10)$ lies on $C$ . The tangent to $C$ at $P$ meets the $y$ -axis at the point $Q$ .

The finite region $R$ , shown shaded in Figure 1, is bounded by $C$ , the $y$ -axis and the line segment $PQ$ .
(b) Use algebraic integration to find the exact area of $R$ .

(8 marks)
Level 5 · 12 marksTry it
3
The curve $C$ has equation $y = 4x - x^2$ and the line $l$ has equation $y = x$ . The region $R$ is bounded by $C$ and $l$ , as shown.(a) Show that $C$ and $l$ intersect where $x = 0$ and $x = 3$ . (3 marks)
(b) Find the area of $R$ . (5 marks)
Level 5 · 8 marksTry it
4
The curve $C$ has equation $y = 6x - x^2$ and the line $l$ has equation $y = 5$ . The region $R$ is enclosed between $C$ and $l$ .(a) Show that $C$ meets $l$ where $x = 1$ and $x = 5$ . (3 marks)
(b) Find the area of $R$ . (5 marks)
Level 5 · 8 marksTry it
5
The region $R$ is bounded by the curve $y = \sqrt{x}$ , the $x$ -axis and the line $x = 4$ .
(a) Find the area of $R$ . (3 marks)
(b) The line $x = a$ divides $R$ into two regions of equal area. Find the value of $a$ , to 3 s.f. (4 marks)
Level 5 · 7 marksTry it

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